Deciding whether to conduct a measurement is a fundamental tenant of quantum physics. A new analysis finds situations in the linear regime where measurement-based feedback control of a quantum system has no merit.
Quantum engineering has seen rapid growth in the past two decades. Physicists, mathematicians and engineers have been working in unison to control a number of diverse systems in the quantum regime. Quantum control involving feedback has become particularly topical, as using information gained from a system can lead to more stable operation of a control protocol. Quantum feedback can be split into two paradigms: measurement based feedback and coherent feedback. Although it is increasingly evident that retaining the coherence of the feedback signal provides an intrinsic advantage over measurement-based feedback, coherent feedback is still a new paradigm. In particular, there are a limited number of options for coherently estimating a state within a feedback loop. As a step towards better understanding and implementation of quantum feedback, this thesis reports on modelling, estimation and control of quantum systems. The first topic is physical realisability of a variety of quantum systems, especially finite level systems and their outgrowths. Bilinear quantum stochastic differential equations (QSDEs) have to be employed to characterise these systems, which is a significant complement to the previous work on linear QSDEs. In light of the fact that direct and indirect couplings play a vital role in quantum network and control, we also provide state-space models for different classes of coupled open quantum systems incorporating both bidirectional and directional interactions. The second topic is coherent observers and optimal filtering. It is well established that classically estimation using the Kalman filter can provide improved performance over direct feedback schemes, and similar demonstrations have been performed for measurement-based quantum feedback. In this thesis I present coherent observers, including least mean squares estimators, which are driven by the coherent output of a specified quantum plant and designed such that some subset of the observer and plant's expectation values converge in the asymptotic limit. Not only can estimators "observe" mean quantities of quantum plant, but also they can "estimate" quantum correlations such as entanglement. This is of much importance to coherent feedback design in the absence of measurement steps without loss of fidelity. Last but not least, several topics regarding stabilisation and control design of quantum systems are discussed in my thesis. To be specific, tools for quantum stability analysis (e.g. Lyapunov conditions) are provided. Both measurement-based feedback and coherent feedback control design are taken into account. A measurement-based optimal controller for opto-mechanical systems aimed at synchronising different mechanical modes is presented, which is application-oriented. Moreover, A general observer-based coherent feedback control framework is studied, and I demonstrate a pole-placement approach via coherent observers that can be applied to various scenarios.
Recent theoretical and experimental advances mean that it is now possible to control physical systems at the quantum level. Indeed, developments in quantum technology provide strong motivation for the feedback control of quantum systems. This paper will discuss both measurement feedback and coherent feedback control for quantum systems. Measurement feedback involves conditioning on the measurement information, and so we describe the Belavkin quantum filter and its use in quantum control. In contrast, coherent feedback does not involve any measurements, and we discuss some recent results on coherent feedback control.
Progress in quantum physics has made it possible to perform experiments in which individual quantum systems are monitored and manipulated in real time. The advent of such new technical capabilities provides strong motivation for the development of theoretical and experimental methodologies for quantum feedback control. The availability of such methods would enable radically new approaches to experimental physics in the quantum realm. Likewise, the investigation of quantum feedback control will introduce crucial new considerations to control theory, such as the uniquely quantum phenomena of entanglement and measurement back-action. The extension of established analysis techniques from control theory into the quantum domain may also provide new insight into the dynamics of complex quantum systems. We anticipate that the successful formulation of an input-output approach to the analysis and reduction of large quantum systems could have very general applications in nonequilibrium quantum statistical mechanics and in the nascent field of quantum information theory.
Quantum systems exhibit unique behaviors governed by principles distinct from classical physics, notably in the realms of measurement and control. This paper explores the fundamental concepts and applications of quantum measurement and feedback control in quantum systems. Measurement in quantum mechanics disrupts the state of a system due to the inherent probabilistic nature of quantum states. This disruption, often described by the collapse of the wave function, presents challenges and opportunities for understanding and manipulating quantum states. Quantum measurement theory addresses how observations affect quantum states and the role of measurement devices in this process. Feedback control mechanisms offer a pathway to mitigate the disruptive effects of measurement and harness them for coherent quantum operations. By continuously monitoring and adjusting quantum systems based on measurement outcomes, feedback control enables stabilization, state preparation, and enhanced coherence times. Examples include quantum error correction, quantum state tomography, and real-time manipulation of quantum bits (qubits).
Abstract : During the three-year performance period we have carried out a program of theoretical research on the foundations and potential applications of coherent-feedback quantum control. We have focused on potential applications in quantum-enhanced meterology and information processing, and on hardware platforms of an optical or opto-mechanical nature. The studies we have been performing begin to build a picture of how coherent feedback can provide a kind of circuit/network theory for quantum engineering, enabling rigorous analysis and numerical simulation of advanced devices and systems. We have begun to establish analytical approaches for comparing the performance of quantum-enhanced feedback control systems with classical counterparts, leveraging canonical results from classical control theory together with advanced methods for numerical simulations of proposed quantum systems. All of our results point to the interest of continued future study in this area, and of future investments in the experimental realization of the kinds of systems we propose.